**2. Basic Governing Equations**

A MHD mixed convection flow of three types of metals (Al, Ag, Cu) in a host Casson fluid over an isothermal sphere of radius *a* with a prescribed wall l temperature *Tw* and ambient l temperature *T*∞ were taken into account. Additionally, a heated and cooled sphere (*Tw* > *T*∞ & *Tw* < *T*<sup>∞</sup>, respectively) was considered.

Figure 1 depicts the schematic configuration and geometrical coordinates, where *U*<sup>∞</sup>, and *g* are the free stream velocity, and the gravity vector, respectively. The (ξ 0 , 0 η) coordinates were measured along the circumference of the sphere at the stagnation point (ξ 0 ≈ 0), and the distance normal to the surface of the sphere, respectively.

**Figure 1.** Schematic configuration of the problem.

Based on the previous assumption, the governing PDEs. for the Casson nanofluid are:

$$\frac{\partial}{\partial \overline{\xi}} (r\overline{u}) + \frac{\partial}{\partial \overline{\eta}} (r\overline{v}) = 0,\tag{1}$$

$$\sqrt{\mu}\frac{\partial\overline{\boldsymbol{u}}}{\partial\overline{\boldsymbol{\xi}}} + \boldsymbol{v}\frac{\partial\overline{\boldsymbol{u}}}{\partial\overline{\boldsymbol{\eta}}} = \tilde{\boldsymbol{u}}\_{\varepsilon}\frac{d\overline{\boldsymbol{u}}\_{\varepsilon}}{d\overline{\boldsymbol{\xi}}} + \boldsymbol{v}\_{nf}\bigg(1 + \frac{1}{\beta}\bigg)\frac{\partial^{2}\overline{\boldsymbol{u}}}{\partial\overline{\boldsymbol{\eta}^{2}}} + \left(\frac{\chi\rho\_{s}\beta\_{s} + (1-\chi)\rho\_{f}\beta\_{f}}{\rho\_{nf}}\right)\mathbf{g}(T - T\_{\rm{ov}})\sin\left(\frac{\overline{\boldsymbol{\xi}}}{\overline{\boldsymbol{a}}}\right) - \frac{\sigma\_{nf}B\_{0}^{2}}{\rho\_{nf}}\overline{\boldsymbol{u}}\_{\varepsilon} \tag{2}$$

$$
\overline{u}\frac{\partial T}{\partial \overline{\xi}} + \overline{v}\frac{\partial T}{\partial \overline{\eta}} = \alpha\_{nf}\frac{\partial^2 T}{\partial \overline{\eta}^2},\tag{3}
$$

When they are associated with the boundary conditions:

$$
\widetilde{\boldsymbol{\mu}} = \widetilde{\boldsymbol{\upsilon}} = \boldsymbol{0}, \ T = T\_{\overline{\boldsymbol{w}}}, \text{ as } \widetilde{\boldsymbol{\eta}} = \boldsymbol{0}, \widetilde{\boldsymbol{\mu}} \to \widetilde{\boldsymbol{u}}\_{\mathcal{E}}(\overline{\boldsymbol{\xi}}), \ T \to T\_{\otimes \ast} \quad \text{as } \widetilde{\boldsymbol{\eta}} \to \infty. \tag{4}
$$

where 0*r*(ξ ) and 0*ue*(ξ 0 ) are given by:

0

$$
\widetilde{r}(\overline{\xi}) = a \sin(\overline{\xi}/a), \text{and } \widetilde{u}\_{\varepsilon}(\overline{\xi}) = \frac{3}{2} l l\_{\infty} \sin(\overline{\xi}/a), \tag{5}
$$

The properties of the nanofluid (defined by [45]) are:

$$\begin{aligned} \frac{\sigma\_{\text{nf}}}{\sigma\_{f}} &= 1 + \frac{3(\sigma - 1)\_{\text{X}}}{(\sigma + 2) - (\sigma - 1)\chi'}, \; \sigma = \frac{\varrho\_{\text{s}}}{\sigma\_{f}}, \; \frac{k\_{\text{nf}}}{k\_{f}} = \frac{\left(k\_{\text{s}} + 2k\_{f}\right) - 2\chi\left(k\_{f} - k\_{\text{s}}\right)}{\left(k\_{\text{s}} + 2k\_{f}\right) + \chi\left(k\_{f} - k\_{\text{s}}\right)}, \; \mu\_{\text{nf}} &= \frac{\mu\_{f}}{\left(1 - \chi\right)^{2.5}}, \\ \left(\rho c\_{p}\right)\_{\text{nf}} &= \left(1 - \chi\right) \left(\rho c\_{p}\right)\_{f} + \chi\left(\rho c\_{p}\right)\_{\text{s}'} \; \rho\_{\text{nf}} = \left(1 - \chi\right)\rho\_{f} + \chi\rho\_{\text{s}\*} \; a\_{\text{nf}} = \frac{k\_{\text{nf}}}{\left(\rho c\_{p}\right)\_{\text{nf}}}, \end{aligned} \tag{6}$$

The following non-dimensional variables that are expressed by Rashad et al. [46] were used:

$$x = \frac{\overleftarrow{\xi}}{a}, \ y = \operatorname{Re}^{1/2}\left(\frac{\overleftarrow{\eta}}{a}\right), r(\overleftarrow{\xi}) = \frac{\overleftarrow{r}(\overleftarrow{\xi})}{a}, \ u = \frac{\overleftarrow{u}}{\operatorname{LI}\_{\text{ov}}},$$

$$v = \operatorname{Re}^{1/2}\left(\frac{\overleftarrow{v}}{\operatorname{LI}\_{\text{ov}}}\right), u\_{\varepsilon}(\xi) = \frac{\overline{u}\_{\varepsilon}(\overleftarrow{\xi})}{\operatorname{LI}\_{\text{ov}}}, \ \theta = \frac{T - T\_{\text{ov}}}{T\_{w} - T\_{\text{ov}}},\tag{7}$$

where Re = *U*∞ *avf*is the Reynolds number.

By substituting Equation (7) into Equations (1)–(4) we ge<sup>t</sup> the following non-dimensional equations:

$$\frac{\partial}{\partial \xi}(ru) + \frac{\partial}{\partial \eta}(rv) = 0,\tag{8}$$

$$\begin{split} \mu \frac{\partial u}{\partial \boldsymbol{\zeta}} + \upsilon \frac{\partial u}{\partial \boldsymbol{\eta}} &= u\_{\mathfrak{e}}(\boldsymbol{\xi}) \frac{d u\_{\mathfrak{e}}}{d \boldsymbol{\xi}} + \frac{\rho\_f}{\rho\_{nf}} \frac{1}{(1 - \chi)^{2.5}} \Big( 1 + \frac{1}{\beta} \right) \frac{\partial^2 u}{\partial \boldsymbol{\eta}^2} \\ &+ \left( \frac{\chi \rho\_s \mathbb{A}\_s + (1 - \chi) \rho\_f \beta\_f}{\rho\_{nf}} \right) \lambda \boldsymbol{\mathcal{O}} \sin \boldsymbol{\xi} - \frac{\rho\_f \sigma\_{nf}}{\rho\_{nf} \sigma\_f} \boldsymbol{M} \boldsymbol{u}\_{\boldsymbol{\prime}} \end{split} \tag{9}$$

$$u\frac{\partial\theta}{\partial\xi} + v\frac{\partial\theta}{\partial\eta} = \frac{1}{\text{Pr}} \left(\frac{k\_{nf}/k\_f}{(1-\chi) + \chi\{\rho\varepsilon\_p\}\_s/\{\rho\varepsilon\_p\}\_f}\right) \frac{\partial^2\theta}{\partial\eta^2} \tag{10}$$

here *M* = <sup>σ</sup>*f* β20*a* ρ*f vf* , Pr = *vf* <sup>α</sup>*f* , λ = *Gr*/Re2, and *Gr* = *<sup>g</sup>*β*f*(*Tw* − *<sup>T</sup>*∞) *a*3 ν2*f* and the dimensionless boundary conditions are:

$$u = v = 0, \ \theta = 1, \text{at } \eta = 0,$$

$$u \to \frac{3}{2} \sin \overline{\xi}, \ \theta \to 0, \ \text{as } \eta \to \infty. \tag{11}$$

To solve the non-dimensional Equations (8)–(10), associated with the boundary conditions in Equation (11), defined the non-dimensional stream function ψ is defined as the following (defined by Nazar et al. [43]):

$$\psi = \overline{\xi} \overline{r}(\overline{\xi}) \overline{F}(\overline{\xi}, \overline{\eta}), \ \theta = \theta(\overline{\xi}, \overline{\eta})\_{\prime}$$

$$u = \frac{1}{r} \frac{\partial \psi}{\partial \overline{\eta}} \text{ and } v = -\frac{1}{r} \frac{\partial \psi}{\partial \overline{\xi}}\tag{12}$$

By using Equation (12), the non-dimensional Equations (8)–(10) are reduced to:

$$\begin{cases} \frac{\rho\_f}{\rho\_{nf}} \frac{1}{(1-\chi)^{2.5}} \Big(1+\frac{1}{\beta}\Big) \frac{\partial^2 F}{\partial \eta^2} + (1+\xi \cot \xi) F \frac{\partial^2 F}{\partial \eta^2} - \left(\frac{\partial F}{\partial \eta}\right)^2 - \frac{\rho\_f \circ \rho\_f}{\rho\_{nf} \circ \rho\_f} M \frac{\partial F}{\partial \eta} \\ + \left(\frac{\chi \rho\_f \theta\_\* / \beta\_f + (1-\chi)\rho\_f}{\rho\_{nf}}\right) \lambda \partial \frac{\sin \xi}{\xi} + \frac{\varrho}{4} \frac{\sin \xi \cos \xi}{\xi} = \xi \Big(\frac{\partial F}{\partial \eta} \frac{\partial^2 F}{\partial \xi \partial \eta} - \frac{\partial F}{\partial \xi} \frac{\partial^2 F}{\partial \eta^2}\Big) \end{cases} \tag{13}$$

$$\frac{1}{\Pr} \left| \frac{k\_{nf}/k\_f}{(1-\chi) + \chi \{\rho c\_p\}\_s/\left(\rho c\_p\right)\_f} \right| \frac{\partial^2 \theta}{\partial \eta^2} + (1 + \xi \cot \xi) F \frac{\partial \theta}{\partial \eta} = \xi \left( \frac{\partial F}{\partial \eta} \frac{\partial \theta}{\partial \xi} - \frac{\partial F}{\partial \xi} \frac{\partial \theta}{\partial \eta} \right), \tag{14}$$

and the boundary conditions become:

$$\frac{\partial F}{\partial \eta} = F = 0, \ \theta = 1 \text{ at } \eta = 0,$$

$$\frac{\partial F}{\partial \eta} \to \frac{3}{2} \frac{\sin \xi}{\xi}, \ \theta \to 0, \ \text{ as } \eta \to \infty. \tag{15}$$

At the stagnation point of the sphere when (ξ ≈ 0), Equations (13)–(15) reduce to:

$$\begin{split} \frac{\rho\_f}{\rho\_{nf}} \frac{1}{(1-\chi)^{2.5}} \Big(1 + \frac{1}{\beta} \Big) \mathbf{F}^{\prime\prime\prime} &+ 2F\mathbf{F}^{\prime\prime} - \left(F^{\prime}\right)^2 - \frac{\rho\_f \sigma\_{nf}}{\rho\_{nf} \sigma\_f} \text{MF}^{\prime} \\ &+ \Big(\frac{\chi \rho\_t \pounds \times / \rho\_f + (1-\chi)\rho\_f}{\rho\_{nf}}\Big) \lambda \theta + \frac{\varrho}{4} = 0, \end{split} \tag{16}$$

$$\frac{1}{\Pr} \left( \frac{k\_{nf} / k\_f}{(1 - \chi) + \chi \{\rho c\_p\}\_s / \{\rho c\_p\}\_f} \right) \theta^{\prime \prime} + 2F\theta^{\prime} = 0,\tag{17}$$

The subject to

$$F' = F = 0, \ \theta = 1 \text{ at } \eta = 0,$$

$$F' \to \frac{3}{2}, \ \theta \to 0, \ \text{ as } \eta \to \infty. \tag{18}$$

In this work two physical quantities were taken into consideration, specifically the local skin friction coefficient *Cf* and the local Nusselt number *Nu*, which are given by Molla et al. [47]:

$$\mathcal{C}\_f = \left(\frac{\tau\_{w\prime}}{\rho l L\_{\infty}^2}\right) Nu = \left(\frac{aq\_w}{k\_f (T\_w - T\_{\infty})}\right) \tag{19}$$

where

$$
\pi\_{\mathfrak{w}} = \mu\_{\mathfrak{n}f} \left( \frac{\partial \widetilde{\boldsymbol{u}}}{\partial \overline{\boldsymbol{\eta}}} \right)\_{\widetilde{\boldsymbol{\eta}}=\boldsymbol{0}} \boldsymbol{q}\_{\mathfrak{w}} = -k\_{\mathfrak{n}f} \left( \frac{\partial \boldsymbol{T}}{\partial \overline{\boldsymbol{\eta}}} \right)\_{\widetilde{\boldsymbol{\eta}}=\boldsymbol{0}}.\tag{20}
$$

Using Equations (7) and (11), *Cf* and *Nu* are turned into:

$$\mathrm{Re}^{1/2}\mathbb{C}\_f = \frac{1}{\left(1-\chi\right)^{2.5}} \Big(1+\frac{1}{\beta}\Big) \xi \frac{\partial^2 F}{\partial \eta^2} (\xi, 0), \\ \mathrm{Re}^{-1/2}\mathrm{Nu} = \frac{-k\_{\mathrm{nf}}}{k\_f} \Big(\frac{\partial \theta}{\partial \eta}\Big)\_{\eta=0}. \tag{21}$$
